Overview. Terrestrial planets grew in a complex series of late stage giant impacts, and Moon-formation was among last to occur around Earth. But was it a singular event? Here we propose it was three or more episodes involving two bodies and the Sun, an almost-merger followed by an interlude, followed by a merger.
The standard model originated in the 1970s and has been surprisingly resilient . It succeeds at making a Moon-size moon with Moon-like rocky and volatile-depleted composition, and sets the stage for the lunar magma ocean and tidal evolution. The model is only relevant to vimp no faster than ~1.1 vesc, which represents the median velocity of late-stage collisions [3,4]. Larger-velocity collisions encompass the other half of the probability distribution. At one end of the energetic spectrum, Theia could have originated from further out, as proposed by , but these collisions are dynamically disfavored .
To study giant impacts outside the limited realm of efficient mergers, our group has applied machine learning  and physical scaling  to hundreds of high resolution simulations over the expected velocities and impact angles and mass ratios. A definitive conclusion is that hit-and-run collisions (HRCs) are much more probable than mergers in the common velocity range vimp ~1.1−1.4 vesc. They spin up the target, strip mantle from the projectile , and send it back into space along with its remnants − but do not create the Moon (although see ).
The solution we propose (Figure 1) is that Theia is the "runner" from a prior HRC with protoearth. The one-two blow leads to greater mixing and slows the bodies down by tens of percent so they will merge eventually.  found that many or even most late stage accretions are collision chains of varying complexity; on this basis we explore a three-act origin of the Moon: an HRC between prototheia and protoearth; a ~103 to 106 yr interlude featuring sweep-ups and close encounters; and a low-velocity merger resembling the standard model.
The standard model is appealing for astrophysical and petrological reasons. It invokes what seems to be a typical "late stage" accretion of two terrestrial embryos, resulting in an Earth-Moon system of the right mass and angular momentum. And it predicts a volatile-depleted silicate Moon with a small iron core. Also, it is a near-perfect merger of the type that is implicitly assumed in N-body simulations. Under those assumptions runners cannot exist.
To address the major deficiencies of the standard model while preserving its major strengths, we have developed a theoretical basis for a collision chain origin of the Moon. We show it to be a common pathway of planet formation, slowing the random velocities until merger is probable. There are innumerable pathways so it is premature to hone in on one scenario. A scenario meriting further research is an ~0.2 MEarth planet that become a mantle-stripped Theia, that then returns thousands to millions of years later for a merger on a strongly unaligned impact axis.
Modeling. Act I assumes a non-rotating protoearth 0.9 MEarth in circular orbit at 1 AU, of 30/70wt% iron/rock composition. Prototheia is 0.15 MEarth with the same composition and entropy profile. We model representative HRCs with vimp = 1.1 or 1.2 vesc and impact angles θimp = 43° to 55°. These are selected so the runner ends up ~0.1 MEarth to match the standard model. The target neither gains nor loses much mass, but it acquires a rotation period of 8 to 11 hours for these HRCs. No significant disk is produced.
We calculate the egress velocity of the runner, which we then transfer into an N-body code  for Act II. To represent all possible collisions, we clone each SPH outcome into 1000 random orientations and evolve each clone, including the other major planets, until they have another collision with a planet, or for 50 Myr when most giant impact chains are finished. One set of outcomes is shown in Figure 2, where the HRC velocity vimp/vesc = 1.20 (red dashed line) gets slowed down to an egress velocity of 1.01 vesc (black dashed line). Most clones return, tallied in Figure 2 by return velocity and interlude duration. Most return in ~105 years at close to the egress velocity.
By design, Act III resembles the standard model, but with an extra stage of mixing that helps solve the lunar isotope problem and slows the bodies down so that merger is probable. We apply the same compositional and entropy profiles to the target, but give it more mass (0.95 MEarth) and induce a rotation period of 8-12 hours per Act I. Theia, the runner, is now 0.1 MEarth. Some of its mantle comes from protoearth, which would further reconcile the isotopic similarities. Altough every HRC is different, for simplicity we assume one 30/70wt% runner. We set the return velocity as either 1.00 or 1.05 vesc and assume a nominal impact angle of 45°. Another variable is the offset angle between the two collisions, ranging from prograde (0°) to retrograde (180°), which would follow a random distribution . The modeled return collisions are therefore variants of the standard model, and end up with a lunar mass or more in orbit in our simulations.
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